Methods for computing an efficient and accurate numerical solution of the real monic
unilateral quadratic matrix equation,
are few. They are not guaranteed to work on all problems. One of the methods performs a
sequence of Newton iterations until convergence occurs whilst another is a matrix analogy
of the scalar polynomial algorithm. The former fails from a poor starting point and the
latter fails if no dominant solution exists. A recent approach, the Elimination method,
is analysed and shown to work on problems for which other methods fail. . The method
requires the coefficients of the characteristic polynomial of a matrix to be computed and
to this end a comparative numerical analysis of a number of methods for computing the
coefficients is performed. A new minimisation approach for solving the quadratic matrix
equation is proposed and shown to compare very favourably with existing methods .
. A special case of the quadratic matrix equation is the matrix square root problem,
where P = o. There have been a number of method proposed for it's solution, the more
successful ones being based upon Newton iterations or the Schur factorisation. The Elimination
method is used as a basis for generating three methods for solving the matrix square
root problem. By means of a numerical analysis and results it is shown that for small order
problems the Elimination methods compare favourably with the existing methods.
The algebraic Riccati equation of stochastic and optimal control is,
where the solution of interest is the symmetric non-negative definite one. The current
methods are based on Newton iterations or the determination of the invariant subspace of
the associated Hamiltonian matrix. A new method based on a reformulation of Newton's
method is presented. The method reduces the work involved at each iteration by introducing
a Schur factorisation and a sparse linear system solver. Numerical results suggest
that it may compare favourably with well-established methods.
Central to the numerical issues are the discussions on conditioning, stability and accuracy.
For a method to yield accurate results, the problem must be well-conditioned and the
method that solves the problem must be stable-consequently discussions on conditioning
and stability feature heavily in this thesis.
The units of measure we use to compare the speed of the methods are the operations
count and the Central Processor Unit (CPU) time. We show how the CPU time accurately
reflects the amount of work done by an algorithm and that the operations counts of the
algorithms correspond with the respective CPU times.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.